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3.8: Torque

Notation:

• $$\boldsymbol\tau_{C}$$ = vector sum of all the torques about C.
• $$\boldsymbol\tau$$ = vector sum of all the torques about the origin O.
• $$\textbf{F}$$ = vector sum of all the external forces.

Theorem

$\boldsymbol\tau = \boldsymbol\tau_{C} + \overline{\textbf{r}} \times \textbf{F}$

Thus:

\begin{align} \boldsymbol\tau &= \sum \textbf{r}_{i} \times \textbf{F}_{i} = \sum (\textbf{r}'_{i} + \overline{\textbf{r}}) \times \textbf{F}_{i} \\ &= \sum \textbf{r}'_{i} \times \textbf{F}_{i} + \overline{\textbf{r}} \sum \textbf{F}_{i} \end{align}

therefore

$\qquad \boldsymbol\tau = \boldsymbol\tau_{C} +\overline{\textbf{r}} \times \textbf{F}$